Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map g : X --> T, T is a topological space, such that for each t is an element of T the set g(-1)(t) has subexponential growth rate in X and the topological dimension dim T = k is minimal among all such maps. Our main result is the inequality rank(h) X less than or equal to corank X for a large class of metric spaces X including all locally compact Hadamard spaces, where rank(h) X is the maximal topological dimension of partial derivative(infinity)Y among all CAT(-1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rank(h) conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding H-n --> X of the standard hyperbolic space H-n with n - 1 > dim X - rank X.